Question

In the journal Mental Retardation, an article reported the results of a peer tutoring program to help mildly mentally retarded children learn to read. In the experiment, the mildly retarded children were randomly divided into two groups: the experimental group received peer tutoring along with regular instruction, and the control group received regular instruction with no peer tutoring. There were n1 = n2 = 36 children in each group. The Gates-MacGintie Reading Test was given to both groups before instruction began. For the experimental group, the mean score on the vocabulary portion of the test was x1 = 344.5, with sample standard deviation s1 = 49.7. For the control group, the mean score on the same test was x2 = 328.5, with sample standard deviation s2 = 47.7. Use a 1% level of significance to test the hypothesis that there was no difference in the vocabulary scores of the two groups before the instruction began.

(a) What is the level of significance?

State the null and alternate hypotheses.

H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂

H₀: μ₁ = μ₂; H₁: μ₁ > μ₂

H₀: μ₁ = μ₂; H₁: μ₁ < μ₂

H₀: μ₁ ≠ μ₂; H₁: μ₁ = μ₂

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. Both sample sizes are large with known standard deviations.

The Student's t. Both sample sizes are large with unknown standard deviations.

The Student's t. Both sample sizes are large with known standard deviations.

The standard normal. Both sample sizes are large with unknown standard deviations.

What is the value of the sample test statistic? (Test the difference μ₁ − μ₂. Round your answer to three decimal places.)

(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(e) Interpret your conclusion in the context of the application.

Reject the null hypothesis, there is insufficient evidence that there is a difference in mean vocabulary scores between the control and experimental groups.

Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean vocabulary scores between the control and experimental groups.

Reject the null hypothesis, there is sufficient evidence that there is a difference in mean vocabulary scores between the control and experimental groups.

Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean vocabulary scores between the control and experimental groups.

Answer 1

a)

0.01 is the level of significance

H0: μ1 = μ2; H1: μ1 ≠ μ2

b)
The standard normal. Both sample sizes are large with known standard deviations.

test statistics:

t = (x1 -x2)/sqrt(s1^2/n1+s2^2/n2)
= (344.5 - 328.5)/sqrt(49.7^2/36 + 47.7^2/36)
= 1.394

c)

p value = 0.1697

d)

At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

e)

Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean vocabulary scores between the control and experimental groups.